Problem: Find $\sin\left(285^\circ\right)$ exactly using an angle addition or subtraction formula. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\sqrt{2}-\sqrt{6}}{4}$ (Choice B) B $\dfrac{-\sqrt{2}-\sqrt{6}}{4}$ (Choice C) C $\dfrac{1+\sqrt{3}}{2}$ (Choice D) D $-\dfrac{\sqrt{8}}{3}$
Explanation: The strategy First, we should rewrite the given angle $285^\circ$ as the sum or difference of two special angles. Then, we can use the sine addition or subtraction identities in order to evaluate $\sin\left(285^\circ\right)$. [How do we find the trigonometric value of a sum or difference?] Rewriting $285^\circ$ We can rewrite $285^\circ$ as follows. $\begin{aligned}285^\circ&=330^\circ-45^\circ\end{aligned}$ In other words, $285^\circ$ is the difference of the special angles $330^\circ$ and $45^\circ$. Evaluating $\sin\left(285^\circ\right)$ Using the sine subtraction identity, we get the following. $\begin{aligned} \sin\left(285^\circ\right)&= \sin\left(330^\circ-45^\circ\right) \\\\\\ &= \sin \left(330^\circ\right) \cos \left(45^\circ\right) - \cos \left(330^\circ\right) \sin \left(45^\circ\right) \\\\\\ &=\left(-\dfrac{1}{2}\right) \left(\dfrac{\sqrt{2}}{2}\right) - \left(\dfrac{\sqrt{3}}{2}\right) \left(\dfrac{\sqrt{2}}{2}\right) \\\\\\ &=\left(-\dfrac{\sqrt{2}}{4}\right) - \left(\dfrac{\sqrt{6}}{4}\right)\\\\\\ &=\dfrac{-\sqrt{2}-\sqrt{6}}{4} \end{aligned}$ Summary $\sin\left(285^\circ\right) = \dfrac{-\sqrt{2}-\sqrt{6}}{4}$